Hardback : £73.21
In a surprising sequence of developments, the longest increasing subsequence problem, originally mentioned as merely a curious example in a 1961 paper, has proven to have deep connections to many seemingly unrelated branches of mathematics, such as random permutations, random matrices, Young tableaux, and the corner growth model. The detailed and playful study of these connections makes this book suitable as a starting point for a wider exploration of elegant mathematical ideas that are of interest to every mathematician and to many computer scientists, physicists and statisticians. The specific topics covered are the Vershik-Kerov-Logan-Shepp limit shape theorem, the Baik-Deift-Johansson theorem, the Tracy-Widom distribution, and the corner growth process. This exciting body of work, encompassing important advances in probability and combinatorics over the last forty years, is made accessible to a general graduate-level audience for the first time in a highly polished presentation.
Dan Romik is Professor of Mathematics at the University of California, Davis.
1. Longest increasing subsequences in random permutations; 2. The Baik-Deift-Johansson theorem; 3. Erdos-Szekeres permutations and square Young tableaux; 4. The corner growth process: limit shapes; 5. The corner growth process: distributional results; Appendix: Kingman's subadditive ergodic theorem.
Show moreIn a surprising sequence of developments, the longest increasing subsequence problem, originally mentioned as merely a curious example in a 1961 paper, has proven to have deep connections to many seemingly unrelated branches of mathematics, such as random permutations, random matrices, Young tableaux, and the corner growth model. The detailed and playful study of these connections makes this book suitable as a starting point for a wider exploration of elegant mathematical ideas that are of interest to every mathematician and to many computer scientists, physicists and statisticians. The specific topics covered are the Vershik-Kerov-Logan-Shepp limit shape theorem, the Baik-Deift-Johansson theorem, the Tracy-Widom distribution, and the corner growth process. This exciting body of work, encompassing important advances in probability and combinatorics over the last forty years, is made accessible to a general graduate-level audience for the first time in a highly polished presentation.
Dan Romik is Professor of Mathematics at the University of California, Davis.
1. Longest increasing subsequences in random permutations; 2. The Baik-Deift-Johansson theorem; 3. Erdos-Szekeres permutations and square Young tableaux; 4. The corner growth process: limit shapes; 5. The corner growth process: distributional results; Appendix: Kingman's subadditive ergodic theorem.
Show more1. Longest increasing subsequences in random permutations; 2. The Baik–Deift–Johansson theorem; 3. Erdős–Szekeres permutations and square Young tableaux; 4. The corner growth process: limit shapes; 5. The corner growth process: distributional results; Appendix: Kingman's subadditive ergodic theorem.
This book presents for the first time to a general readership recent groundbreaking developments in probability and combinatorics related to the longest increasing subsequence problem.
Dan Romik is Professor of Mathematics at the University of California, Davis.
'The story of longest monotone subsequences in permutations has
been, for six decades, one of the most beautiful in mathematics,
ranging from the very pure to the applied and featuring many
terrific mathematicians, starting with Erdős–Szekeres's 'happy end
theorem' and continuing through the Tracy–Widom distribution and
the breakthrough of Baik–Deift–Johansson. With its connections to
many areas of mathematics, to the Riemann hypothesis, and to
high-energy physics we cannot foresee where the story is heading.
Dan Romik tells the tale thus far - and teaches its rich
multifaceted mathematics, a blend of probability, combinatorics,
analysis, and algebra - in a wonderful way.' Gil Kalai, Hebrew
University of Jerusalem
'How long is the longest increasing subsequence in a random
permutation? This innocent-looking combinatorial problem has
surprisingly rich connections to diverse mathematical areas:
Poisson processes and Last-passage percolation, growth processes
and random matrices, Young diagrams and special functions … Its
solution weaves together some highlights of nineteenth- and
twentieth-century mathematics, yet continues to have growing impact
in the twenty-first. Dan Romik's excellent book makes these
exciting developments available to a much wider mathematical
audience than ever before. The minimal prerequisites ensure that
the reader will also encounter mathematical tools that have stood
the test of time and can be applied to many other concrete
problems. This is a wonderful story of the unity of mathematics,
and Romik's enthusiasm for it shines through.' Yuval Peres,
Principal Researcher, Microsoft
'This is a marvelously readable book that coaches the reader toward
an honest understanding of some of the deepest results of modern
analytic combinatorics. It is written in a friendly but rigorous
way, complete with exercises and historical sidebars. The central
result is the famous Baik-Deift-Johansson theorem that determines
the asymptotic distribution of the length of the longest increasing
subsequence of a random permutation, but many delicious topics are
covered along the way. Anyone who is interested in modern analytic
combinatorics will want to study this book. The time invested will
be well rewarded - both by enjoyment and by the acquisition of a
powerful collection of analytical tools.' Michael Steele,
University of Pennsylvania
'Mathematics books that concentrate on a problem, rather than on a
technique or a subfield, are relatively rare but can be a
wonderfully exciting way to dive into research. Here we have the
felicitous combination of an extraordinarily fascinating and
fruitful problem and a literate tour guide with a terrific eye for
the best proof. More like a detective story than a text, this
elegant volume shows how a single wise question can open whole new
worlds.' Peter Winkler, Dartmouth College
'Timely, authoritative, and unique in its coverage …' D. V.
Feldman, Choice
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